Positive Linear Systems: Theory and Applications / Edition 1

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This volume introduces the reader to the world of positive linear systems, an important and fascinating class of linear systems. The subject matter is divided into three parts, including definitions and basic properties of liner systems, theoretical results, and the study of some classes of positive linear systems relevant in applications.
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Editorial Reviews

SciTech Book News
Explores a class of linear dynamical systems called positive linear systems whose state variables take only non-negative values.
Mathematical Reviews
This book gives an interesting overview of results regarding single-input single-output, time-invariant, finite-dimensional linear positive systems.
Explores a class of linear dynamical systems called positive linear systems whose state variables take only non-negative values. Farina (University of Rome) and Rinaldi (Milan Polytechnic) begin with definitions and the basic properties of positive linear systems. The main conceptual results are then reported<-->stability, spectral characterization of irreducible systems, positivity of equilibria, reachability and observability, realization, minimum phase, and interconnected systems. The book concludes with the study of some classes of positive linear systems of particular relevance in applications, such as the Leontief model used by economists, the Leslie model used by demographers, the Markov chains, the compartmental models, and the queuing systems. Two appendices review linear algebra and linear systems theory. Annotation c. Book News, Inc., Portland, OR (booknews.com)
From the Publisher
"Explores a class of linear dynamical systems called positive linear systems whose state variables take only non-negative values." (SciTech Book News,Vol. 24, No. 4, December 2000)

"The exposition of the topics is consistent and clear. The book is addressed to graduate students, scientists and engineers in control." (Mathematical Reviews, Issue 2001g)

"This book gives an interesting overview of results regarding single-input single-output, time-invariant, finite-dimensional linear poitive systems." (Mathematical Reviews, 2001g:93001)

"There are lots of things to like about this book. In particular, I liked the appendix on element so f linear systems theory.... Then there is the clear enthusiasm of the authors for the subject...useful for self study or as a supplement in a more advanced course..." (SIAM Review, Vol. 43, No. 3)

"Very well-written and well-organized suitable for students who have had a first course in differential equations." (American Mathematical Monthly, January 2002)

"...the authors really succeed in conveying their enthusiasm and the flavor of the subject..." (Zentralblatt Math, Vol.988, No.13, 2002)

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Product Details

Meet the Author

LORENZO FARINA, PhD, is Associate Professor of Modeling and Simulation, Dipartimento di Informatica e Sistemistica, Universita di Roma "La Sapienza," Italy. SERGIO RINALDI, PhD, is Full Professor of Systems Theory, Dipartimento di Elettronica e Informazione, Politecnico di Milano, Italy.
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Table of Contents

Preface ix
Part I Definitions 1
1 Introduction 3
2 Definitions and Conditions of Positivity 7
3 Influence Graphs 17
4 Irreducibility, Excitability, and Transparency 23
Part II Properties 33
5 Stability 35
6 Spectral Characterization of Irreducible Systems 49
7 Positivity of Equilibria 57
8 Reachability and Observability 65
9 Realization 81
10 Minimum Phase 91
11 Interconnected Systems 101
Part III Applications 107
12 Input-Output Analysis 109
13 Age-Structured Population Models 117
14 Markov Chains 131
15 Compartmental Systems 145
16 Queueing Systems 155
Conclusions 167
Annotated Bibliography 169
Bibliography 177
Appendix A Elements of Linear Algebra and Matrix Theory 187
A.1 Real Vectors and Matrices 187
A.2 Vector Spaces 189
A.3 Dimension of a Vector Space 193
A.4 Change of Basis 195
A.5 Linear Transformations and Matrices 196
A.6 Image and Null Space 198
A.7 Invariant Subspaces, Eigenvectors, and Eigenvalues 201
A.8 Jordan Canonical Form 207
A.9 Annihilating Polynomial and Minimal Polynomial 210
A.10 Normed Spaces 212
A.11 Scalar Product and Orthogonality 216
A.12 Adjoint Transformations 221
Appendix B Elements of Linear Systems Theory 225
B.1 Definition of Linear Systems 225
B.2 ARMA Model and Transfer Function 228
B.3 Computation of Transfer Functions and Realization 231
B.4 Interconnected Subsystems and Mason's Formula 234
B.5 Change of Coordinates and Equivalent Systems 237
B.6 Motion, Trajectory, and Equilibrium 238
B.7 Lagrange's Formula and Transition Matrix 241
B.8 Reversibility 244
B.9 Sampled-Data Systems 244
B.10 Internal Stability: Definitions 248
B.11 Eigenvalues and Stability 248
B.12 Tests of Asymptotic Stability 251
B.13 Energy and Stability 256
B.14 Dominant Eigenvalue and Eigenvector 259
B.15 Reachability and Control Law 260
B.16 Observability and State Reconstruction 264
B.17 Decomposition Theorem 268
B.18 Determination of the ARMA Models 272
B.19 Poles and Zeros of the Transfer Function 279
B.20 Poles and Zeros of Interconnected Systems 282
B.21 Impulse Response 286
B.22 Frequency Response 288
B.23 Fourier Transform 293
B.24 Laplace Transform 296
B.25 Z-Transform 298
B.26 Laplace and Z-Transforms and Transfer Functions 300
Index 303
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